As previously mentioned, this example is reminiscent to that of Section 4.2 of [1]} where Ley et. al. discuss comparing Bayesian posteriors for normal models. Similarly to above, they compare a posterior with uninformative uniform prior against a posterior with normal conjugate prior. Their results show a convergence of \(O(n^{-3/2})\)  compared to ours of \(O(n^{-1})\) . This discrepancy in orders is due to the fact that we are not calculating the expectation \(\mathbb {E}[|\alpha (X)\pi _0^{\prime }(X)\tau ^c(X)|]\)  and are instead bounding it above by some enveloping function; thus, giving us a less optimal order bound.
